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2.3: Subtraction of Rational Numbers

Difficulty Level: At Grade Created by: CK-12

In the previous two lessons, you have learned how to find the opposite of a rational number and to add rational numbers. You can use these two concepts to subtract rational numbers. Suppose you want to find the difference of 9 and 12. Symbolically, it would be \begin{align*}9 - 12\end{align*}. Begin by placing a dot at nine and move to the left 12 units.

\begin{align*}9 - 12 = -3\end{align*}

Rule: To subtract a number, add its opposite.

\begin{align*}3 - 5 = 3 + (-5) = -2 && 9 - 16 = 9 + (-16) = -7\end{align*}

A special case of this rule can be written when trying to subtract a negative number.

The Opposite-Opposite Property: For any real numbers \begin{align*}a\end{align*} and \begin{align*}b, \ a-(-b) = a + b\end{align*}.

Example 1: Simplify \begin{align*}-6 - (-13).\end{align*}

Solution: Using the Opposite-Opposite Property, the double negative is rewritten as a positive.

\begin{align*}-6 - (-13) = -6 + 13 = 7\end{align*}

Example 2: Simplify \begin{align*}\frac{5}{6} - \left ( - \frac{1}{18} \right ).\end{align*}

Solution: Begin by using the Opposite-Opposite Property.

\begin{align*}\frac{5}{6} + \frac{1}{18}\end{align*}

Next, create a common denominator: \begin{align*}\frac{5 \times 3}{6 \times 3} + \frac{1}{18} = \frac{15}{18} + \frac{1}{18}.\end{align*}

Add the fractions: \begin{align*}\frac{16}{18}.\end{align*}

Reduce: \begin{align*}\frac{2 \times 2 \times 2 \times 2}{3 \times 3 \times 2} = \frac{8}{9}.\end{align*}

Evaluating Change Using a Variable Expression

You have learned how to graph a function by using an algebraic expression to generate a table of values. Using the table of values you can find the change in the dependent values between any two independent values.

In Lesson 1.5, you wrote an expression to represent the pattern of the total cost to the number of CDs purchased. The table is repeated below:

\begin{align*}&\text{Number of CDs} && 2 && 4 && 6 && 8 && 10\\ &\text{Cost (\$)} && 24 && 48 && 72 && 96 && 120\end{align*}

To determine the change, you must find the difference between the dependent values and divide it by the difference in the independent values.

Example 2: What is the cost of a CD?

Solution: We begin by finding the difference between the cost of two values. For example, the change in cost between 4 CDs and 8 CDs.

\begin{align*}96-48 = 48\end{align*}

Next, we find the difference between the number of CDs.

\begin{align*}&&&8 - 4 = 4\\ &\text{Finally, we divide.}&& \quad \frac{48}{4}=12\end{align*}

Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Subtraction of Rational Numbers (10:22)

In 1 – 20, subtract the following rational numbers. Be sure that your answer is in the simplest form.

  1. \begin{align*}9 - 14\end{align*}
  2. \begin{align*}2 - 7\end{align*}
  3. \begin{align*}21 - 8\end{align*}
  4. \begin{align*}8 - (-14)\end{align*}
  5. \begin{align*}-11 - (-50)\end{align*}
  6. \begin{align*}\frac{5}{12} - \frac{9}{18}\end{align*}
  7. \begin{align*}5.4 - 1.01\end{align*}
  8. \begin{align*}\frac{2}{3} - \frac{1}{4}\end{align*}
  9. \begin{align*}\frac{3}{4} - \frac{1}{3}\end{align*}
  10. \begin{align*}\frac{1}{4} - \left (- \frac{2}{3} \right )\end{align*}
  11. \begin{align*}\frac{15}{11} - \frac{9}{7}\end{align*}
  12. \begin{align*}\frac{2}{13} - \frac{1}{11}\end{align*}
  13. \begin{align*}-\frac{7}{8} - \left (- \frac{8}{3} \right )\end{align*}
  14. \begin{align*}\frac{7}{27} - \frac{9}{39}\end{align*}
  15. \begin{align*}\frac{6}{11} - \frac{3}{22}\end{align*}
  16. \begin{align*}-3.1 - 21.49\end{align*}
  17. \begin{align*}\frac{13}{64} - \frac{7}{40}\end{align*}
  18. \begin{align*}\frac{11}{70} - \frac{11}{30}\end{align*}
  19. \begin{align*}-68 - (-22)\end{align*}
  20. \begin{align*}\frac{1}{3} - \frac{1}{2}\end{align*}
  21. Determine the change in \begin{align*}y\end{align*} between (1, 9) and (5, –14).
  22. Consider the equation \begin{align*}y = 3x + 2\end{align*}. Determine the change in \begin{align*}y\end{align*} between \begin{align*}x = 3\end{align*} and \begin{align*}x = 7.\end{align*}
  23. Consider the equation \begin{align*}y = \frac{2}{3}x + \frac{1}{2}\end{align*}. Determine the change in \begin{align*}y\end{align*} between \begin{align*}x = 1\end{align*} and \begin{align*}x = 2\end{align*}.
  24. True or false? If the statement is false, explain your reasoning. The difference of two numbers is less than each number.
  25. True or false? If the statement is false, explain your reasoning. A number minus its opposite is twice the number.
  26. KMN stock began the day with a price of $4.83 per share. At the closing bell, the price dropped $0.97 per share. What was the closing price of KMN stock?

In 27 – 32, evaluate the expression. Assume \begin{align*}a=2, \ b= -3,\end{align*} and \begin{align*}c = -1.5.\end{align*}

  1. \begin{align*}(a-b)+c\end{align*}
  2. \begin{align*}|b+c|- a\end{align*}
  3. \begin{align*}a-(b+c)\end{align*}
  4. \begin{align*}|b|+ |c|+ a\end{align*}
  5. \begin{align*}7b+4a\end{align*}
  6. \begin{align*}(c-a)- b\end{align*}

Mixed Review

  1. Graph the following ordered pairs: \begin{align*}\left \{(0,0),(4,4),(7,1),(3,8) \right \}\end{align*}. Is the relation a function?
  2. Evaluate the expression when \begin{align*}m= \left (- \frac{2}{3} \right ): \ \frac{2^3+m}{4}\end{align*}.
  3. Translate the following into an algebraic equation: Ricky has twelve more dollars than Stacy. Stacy has 5 less dollars than Aaron. The total of the friends’ money is $62.
  4. Simplify \begin{align*}\frac{1}{3} + \frac{7}{5}\end{align*}.
  5. Simplify \begin{align*}\frac{21}{4} - \frac{2}{3}\end{align*}.

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